Is it permissible to hold a celebration with the intention of bringing joy to a child's heart, and not with the intention of celebrating their birthday, doing so a week or more before or after their birthdate, in opposition to the disbelievers and to avoid imitating them?

Celebrating birthdays is prohibited, and holding the celebration before or after the actual birthday does not change the ruling, as long as the motive is the birth itself. This is because what occurs before or after the Eid (holiday) due to its activities carries the same ruling as the Eid itself.